There are quite a few symbols in this short story. First, the family, mountain folk, live in the Notch, on a rock face. The mountain and the rock, at the close of the story, are symbolic not only of the power of nature, but also become, ironically, the monument, the grave, and the headstones for the entire family and the Ambitious Guest. The eldest daughter is named Happiness and the old grandmother is Happiness grown old, symbolic of the simplicity and peace present in the house and the women folk prior to the ambitious wishes brought on by the guest. A heavy footstep is heard outside the door, symbolic of Death that will shortly beckon all those within the house. The fire, fed with heart's ease, metaphorically, is the only place the family is safe and comforted, yet the family flees when The Slide arrives.
Literally, the slide is the falling rock which will take them to their death. Metaphorically (symbolically), The Slide is their slide from grace, from their peaceful existence to one of hopes and grandeur that could only lead to their spiritual downfall as they are simple, content folk unlike their guest. Ironically the young, refined and educated guest gets what he wishes for - a monument over him at his demise.
Yet, paradoxically, when the Ambitious Guest says, "Then let Death come! I shall have built my monument," he does not realize death is so near nor will he ever be remembered or even known for the accomplishments he wishes to achieve!